For tax purposes one of you is the custodial parent. Here is rule for determining the custodial parent according to the IRS if child stayed with each parent an equal number of nights. For more details and other situations see Publication 504.
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"Example 3—child lived same number of days with each parent.
Your son lived with you 180 nights during the year and lived the same number of nights with his other parent, your ex-spouse. Your adjusted gross income is $40,000. Your ex-spouse's adjusted gross income is $25,000. You are treated as your son's custodial parent because you have the higher adjusted gross income."
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